Also useful against sword characters since you otherwise have no disjoint.
Falco's Dtilt is a (natural) disjoint since it uses his tail. Same deal with Fox's Dtilt and Falco's kept the same tail sweep in every game so far. The only difference is that in every game, but Smash 4, its range was simply Falco's tail length which sometimes stretches in some of the games. So, if you're coming from Smash 4, don't expect to have that fraudulent, even more disjointed, and invisible range when using Dtilt.
Does Falco have any moves that cross-up shields?
If you mean moves that let him pass through shields, then only dash attack I think. Fox and Falco's Side Specials no longer pass through shields, so they can't use them to cross up anymore. Not sure about their Up Specials, but Fox's and Falco's Up Specials take 43 frames to launch, so your opponent has to be very oblivious to or confused as to what's happening to not punish that.
If you mean moves where Falco can quickly hit someone from behind as they're shielding, then jump over them and use Bair or Uair which hits behind him first or turn before jumping and do Nair, Fair, Uair, or Dair. As noted already, Nair's loop hits probably don't have enough shield stun and Fair has the highest landing lag, 15 frames, out of all his aerials and doesn't auto-cancel, so you're going to have to rely on your opponent dropping their shield and getting caught by any of its hits including the landing hit that people still seem to forget exists.
Would you say that fair or nair is a better tool in neutral? I assumed nair because it has less landing lag.
In my opinion, Fair's 15 landing frames and not be able to auto-cancel from a hop makes it too risky and people are relying on its landing hit to save them from being punished. Its landing hit no longer being that disjointed and more like launch Smash 4's landing hit is going to make it even easier for people to punish it if they weren't already shielding all its hits in the first place.
Nair at least has low landing lag at 9 frames even though it doesn't auto-cancel from a hop, but it has two problems and that is shield stun and connecting. Nair being a multi-hit move means its loop hits, hits 1 to 3, are not going to do a lot of damage, so it probably won't cause a lot of shield stun. Most multi-hit moves also don't have finishers that do a lot of damage either even though most of them have high knockback growth. So, if you're relying on trying to inflict as much shield stun as possible, Nair and Fair aren't good options and Nair doesn't have a landing hit. Ultimate Nair seems to be more like Brawl and launch Smash 4 Nair where its loop hits don't have autolink angles, or it functions differently, but I think it's more likely that it doesn't have autolink angles making it easier to drop people compared to 1.0.8 Smash 4 Nair. Autolink angles make it so that the opponent moves with the move they're being hit by and there are different types of auto-link angles. You can read more about it here:
https://www.ssbwiki.com/Autolink_angle.
Nair still works as a fast option with low landing lag, but it not being as reliable as 1.0.8 Smash 4 Nair is a problem. On the bright side, you can use landing Nair and Fair to confirm into other moves. Just avoid landing with their final hit and Fair's landing hit.
Anyway, for aerials to use in neutral, Uair and Bair would probably be safer. Both of them auto-cancel from a hop, both of them are single hit moves, so they will deal more damage and shield stun, and their range aren't as centered on Falco's body like Nair and Fair are. The caveat to using them, however, is that you need to get good at doing aerials from a pivot jump or RAR (reverse aerial rush). People call it reverse aerial rush or RAR, but I find it unnecessarily clunky, so I call it pivot jumping because you run, pivot/turn around, and jump.
On the ground, jab and tilts which you should abuse Ultimate's dash cancel mechanic to use them from his dash and run in addition from his walk, empty hops, and empty jumps, or however else you find yourself able to move with him. Use Blaster as well to control space, especially against characters who are much slower than Falco since if you find it dangerous to approach them or know it's difficult for them to approach Falco, then why approach when you can just tack on damage and annoy them? Punish their bad approaches or calls as they try to find their way around Falco's lasers and other moves.
I don't think Marcbri is a Falco main for Ultimate, but he occasionally play Falco and incorporates Blaster into his game more frequently than other Falco players I've seen so far.
Here's a match of his Falco vs. Jamiro's Mega Man:
https://www.youtube.com/watch?v=kbJ4ToiJI4o.
Here's one against Pocafeiner's Villager and Jigglypuff:
https://youtu.be/UOw-oESLjfg?t=291. The first match he used Cloud and lost which led to him switching to Falco for the rest of the game.
In general, though, before figuring out what moves you should use with what character for neutral and by extension, what combos, tech, and so on, learning how to move well is the first thing I would advise you to do. Good movement means you have control of your character and allows you to be more deceptive. Make use of empty hops and empty jumps that you can mix in with safe aerials like Uair and Bair. Dash and walk in addition to running. Being able to use any move out of dashes and runs are great and all, but Falco's walk speed is fast and you can do anything out of a walk as well.
Edit: conso's Falco looks like good one to study. He should use lasers a bit more against characters like Peach where approaching her is dangerous. Trying to anti-air her might be difficult with anything, but Up Smash. Up Smash out of shield could work if you don't get caught by Peach's aerials. Dtilt which I feel like he should try to use more might be able to catch low floats and bad landings. Peach looks nuts in this game.
conso vs. Weeb's Snake (Winners Semis):
https://www.youtube.com/watch?v=BS2gbT3Wty0.
conso's Falco and Dr. Mario vs. Linoone's Bayonetta and Peach (Winners Semis):
https://www.youtube.com/watch?v=5Ue8dappwL4. So, conso gets overrun by Linoone's Peach who looks like that's his main, but Linoone wanted to play Bayonetta for this tournament.
conso vs. Seymour Butts's Inkling (Losers Quarters):
https://www.youtube.com/watch?v=-57kzPZg5Zg.
conso vs. Bigfateli's Jigglypuff and Fox (Losers Semis):
https://www.youtube.com/watch?v=gBWyWy4IZpA.
conso vs. Linoone's Peach and Daisy (Losers Finals):
https://www.youtube.com/watch?v=dwGyiCm0520. He has a stronger standing against Linoone's Peach and Daisy here than during their previous match.